Figure REF  shows the resulting relation between C\(^+\)  and SFR as measured from the ALPINE sample at \(z=\)  4–6 from [1]}.
The best-fit relation is shown as a solid black line and is derived using the Bayesian fitting method described in [2]}The method is implemented in the Python package linmix at https://github.com/jmeyers314/linmix, accessed on May 31, 2022., which properly treats errors and nondetections in astronomical data during the fitting process.
We also show the location of the stacked robust nondetections from (purple empty star [3]}), which agrees well with the C\(^+\)  vs. SFR relation found by the stacked detections (purple filled stars).
In addition, the observed C\(^+\)  vs. SFR relation of local galaxies (gray line and region (\(2\sigma \) ); [4]}) as well as theoretical models [5]}Note that all of these relations have been calibrated to the same IMF. are also shown. While other studies found C\(^+\)  to be underluminous [6]}, [7]}, [8]}, [9]}, [10]}, the ALPINE sample shows a very consistent relation between C\(^+\)  and total star formation, similar to findings at lower redshifts [4]}. Even more so, the comparison of ALPINE data with measurement from the literature at higher redshifts [12]}, [13]} suggests only a slow (if any) evolution of the C\(^+\)\(-\) SFR relation out to higher redshifts.
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